WASHINGTON, Jan 12 (Reuters) - U.S. Senator Marco Rubio, a Republican presidential candidate, and a fellow Cuban-American lawmaker introduced legislation on Tuesday that would cut back on some of the special benefits given to immigrants from Cuba.

Rubio introduced the bill in the Senate and has made his Cuban heritage a theme of his campaign for his party's nomination to run for the White House in the November election. But the Florida senator has also questioned the generous benefits given to immigrants from the Communist-ruled island.

Republican Representative Carlos Curbelo, also of Florida, introduced the measure in the House of Representatives.

President Barack Obama, a Democrat, and Cuban President Raul Castro announced in December 2014 they would seek more normal relations. In July, diplomatic ties were restored after 54 years of enmity.

MT PLEASANT, SC - JANUARY 13: Republican presidential candidate Sen. Marco Rubio (R-FL) greets guests during a campaign rally at the Water Dog Grill on January 13, 2016 in Mount Pleasant, South Carolina. Tomorrow Rubio will join other candidates seeking the Republican nomination for president for a debate at the North Charleston Coliseum and Performing Arts Center in North Charleston, S.C.. (Photo by Scott Olson/Getty Images)

Senator Marco Rubio, a Republican from Florida and 2016 presidential candidate, waits to speak during a town hall meeting at the Maytag Innovation Center in Newton, Iowa, U.S., on Wednesday, Dec. 30, 2015. Rubio found himself under siege on two fronts Tuesday over his attendance record in the U.S. Senate as the Florida lawmaker embarked on a swing through snowy Iowa. Photographer: Scott Morgan/Bloomberg via Getty Images

MIAMI BEACH, FL - NOVEMBER 15: Republican presidential candidate Sen. Marco Rubio (R-FL) greets people before speaking during a community rally for 'Never Again' which was bringing attention to what the organizers say is a rise in worldwide anti-Semitism and the campaign against Israel's right to exist on November 15, 2015 in Miami Beach, Florida. Rubio continues to campaign for the Republican party's nomination. (Photo by Joe Raedle/Getty Images)

CEDAR RAPIDS, IA - NOVEMBER 21: Republican presidential candidate Sen. Marco Rubio (R-FL) speaks to guests during a town hall meeting on November 21, 2015 in Cedar Rapids, Iowa. Yesterday Rubio participated in the Presidential Family Forum in Des Moines with six of his Republican rivals for the nomination. Rubio has several campaign stops scheduled in the state today. (Photo by Scott Olson/Getty Images)

ORLANDO, FL - NOVEMBER 12: Sen. Marco Rubio (R-FL) speaks at the Sunshine Summit opening dinner at Disney's Contemporary Resort on November 12, 2015 in Orlando, Florida.The dinner is the kick-off of a three-day event that will draw thousands of Republicans, mostly to hear live speeches from all the GOP presidential candidates on Friday and Saturday. (Photo by Tom Benitez - Pool/Getty Images)

ORLANDO, FL - NOVEMBER 13: Republican presidential candidate Sen. Marco Rubio (R-FL) speaks during the Sunshine Summit conference being held at the Rosen Shingle Creek on November 13, 2015 in Orlando, Florida. The summit brought Republican presidential candidates in front of the Republican voters. (Photo by Joe Raedle/Getty Images)

CONCORD, NH - NOVEMBER 05: Republican Presidential candidate Marco Rubio (R-FL) files paperwork for the New Hampshire primary at the State House on November 5, 2015 in Concord, New Hampshire. Each candidate must file paperwork to be on the New Hampshire primary ballot, which will be held February 9, 2016. (Photo by Scott Eisen/Getty Images)

Republican Presidential hopeful Marco Rubio leaves the New Hampshire State House after filing for the state ballot November 5, 2015 in Concord, New Hampshire. AFP PHOTO/DON EMMERT (Photo credit should read DON EMMERT/AFP/Getty Images)

MANCHESTER, NH - NOVEMBER 4: Republican Presidential candidate Marco Rubio (R-FL) participates in a round table discussion at Saint Anselm College in Manchester, New Hampshire. Rubio is looking for a bump in the polls following a strong outing in the last debate. (Photo by Darren McCollester/Getty Images)

GREENVILLE, SC - SEPTEMBER 18: Sen. Marco Rubio (R-FL) speaks to voters at the Heritage Action Presidential Candidate Forum September 18, 2015 in Greenville, South Carolina. Eleven republican candidates each had twenty five minutes to talk to voters Friday at the Bons Secours Wellness arena in the upstate of South Carolina. (Photo by Sean Rayford/Getty Images)

NEW YORK, NY - OCTOBER 06: Republican presidential candidate, Sen. Marco Rubio (R-FL) speaks at Civic Hall about the 'sharing economy' on October 6, 2015 in New York City. Rubio, who has been experiencing a slight uptick in the polls after strong debate performances, has a second book out in paperback this Tuesday called American Dream: Restoring Economic Opportunity for Everyone. (Photo by Spencer Platt/Getty Images)

DES MOINES, IA - AUGUST 18: Republican presidential candidate and U.S. Sen. Marco Rubio (L) (R-FL) mans the grill with U.S. Rep. David Young (R) (R-IA) at the Iowa Pork Producers Pork Tent during the Iowa State Fair on August 18, 2015 in Des Moines, Iowa. Presidential candidates are addressing attendees at the Iowa State Fair on the Des Moines Register Presidential Soapbox stage and touring the fairgrounds. The State Fair runs through August 23. (Photo by Justin Sullivan/Getty Images)

NEW YORK, NY - AUGUST 14: Marco Rubio gives speach for the Foreign Policy Initiative at 3 West Club on August 14, 2015 in New York City. (Photo by Steve Sands/WireImage)

CLEVELAND, OH - AUGUST 06: Republican presidential candidate Sen. Marco Rubio (R-FL) participates in the first prime-time presidential debate hosted by FOX News and Facebook at the Quicken Loans Arena August 6, 2015 in Cleveland, Ohio. The top-ten GOP candidates were selected to participate in the debate based on their rank in an average of the five most recent national political polls. (Photo by Chip Somodevilla/Getty Images)

CLEVELAND, OH - AUGUST 05: Republican presidential candidate Sen. Marco Rubio (R-FL) greets guests gathered for a campaign event at Town Hall on August 5, 2015 in Cleveland, Ohio. Rubio is in Cleveland to participate in the Fox News GOP presidential candidate debate scheduled to take place tomorrow evening. The top ten polling Republican candidates were chosen to participate in the debate. (Photo by Scott Olson/Getty Images)

Senator Marco Rubio, a Republican from Florida and 2016 presidential candidate, speaks during the Faith and Freedom Coalition's 'Road to Majority' legislative luncheon in Washington, D.C., U.S., on Thursday, June 18, 2015. The annual Faith & Freedom Coalition Policy Conference gives top-tier presidential contenders as well as long shots a chance to compete for the large evangelical Christian base in the crowded Republican primary contest. Photographer: Andrew Harrer/Bloomberg via Getty Images

BOONE, IA - JUNE 06: Republican presidential hopeful U.S. Sen. Marco Rubio (R-FL) waits his turn to speak at a Roast and Ride event hosted by freshman Senator Joni Ernst (R-IA) on June 6, 2015 in Boone, Iowa. Ernst is hoping the event, which featured a motorcycle tour, a pig roast, and speeches from several 2016 presidential hopefuls, becomes an Iowa Republican tradition. (Photo by Scott Olson/Getty Images)

GREENVILLE, SC - MAY 09: Republican Presidential candidate U.S. Sen. Marco Rubio (R-FL) greets supporters at the Freedom Summit on May 9, 2015 in Greenville, South Carolina. Rubio joined eleven other potential candidates in addressing the event hosted by conservative group Citizens United. (Photo by Richard Ellis/Getty Images)

NASHUA, NH - APRIL 17: Republican presidential candidate Sen. Marco Rubio (R-FL) speaks at the First in the Nation Republican Leadership Summit April 17, 2015 in Nashua, New Hampshire. The Summit brought together local and national Republicans and was attended by all the Republicans candidates as well as those eyeing a run for the nomination. (Photo by Darren McCollester/Getty Images)

MIAMI, FL - APRIL 13: Republican Senator Marco Rubio of Florida makes a formal announcement that he is entering the 2016 Presidential race at the Freedom Tower on April 13, 2015 in Miami, Florida. (Photos by Charles Ommanney for the Washington Post)

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Rubio and Curbelo are staunch opponents of the policy shift, saying Cuba's government must protect human rights for its citizens before benefiting from closer U.S. ties.

Immigration has been a dominant theme of the presidential campaign, with Republicans in particular calling for strict controls on those who cross U.S. borders.

Benefiting from policies that originated during the Cold War to help those escaping Communist rule, Cubans who reach U.S. soil are automatically given federal benefits, even if they travel back to Cuba. They are also allowed to remain in the country even if they arrive illegally.

Rubio's bill would end the automatic eligibility for federal aid given to Cuban nationals, except for those fleeing persecution.

"It is particularly outrageous when individuals who claim to be fleeing repression in Cuba are welcomed and allowed to collect federal assistance based on their plight, only to return often to the very place they claimed to be fleeing," Rubio said in a statement.

U.S. officials have seen a sharp increase in immigration from Cuba in recent months, as the island's citizens worry that warming relations will end their special status. Thousands were stranded in Costa Rica in December as they tried to reach the southern U.S. border.